A spatial autoregressive multinomial probit model for anticipating land-use change in Austin, Texas

Abstract

This paper develops an estimation strategy for and then applies a spatial autoregressive multinomial probit model to account for both spatial clustering and cross-alternative correlation. Estimation is achieved using Bayesian techniques with Gibbs and the generalized direct sampling (GDS). The model is applied to analyze land development decisions for undeveloped parcels over a 6-year period in Austin, Texas. Results suggest that GDS is a useful method for uncovering parameters whose draws may otherwise fail to converge using standard Metropolis-Hastings algorithms. Estimation results suggest that residential and commercial/civic development tends to favor more regularly shaped and smaller parcels, which may be related to parcel conversion costs and aesthetics. Longer distances to Austin’s central business district increase the likelihood of residential development, while reducing that of commercial/civic and office/industrial uses. Everything else constant, distances to a parcel’s nearest minor, and major arterial roads are estimated to increase development likelihood of commercial/civic and office/industry uses, perhaps because such development is more common in less densely developed locations (as proxied by fewer arterials). As expected, added soil slope is estimated to be negatively associated with residential development, but positively associated with commercial/civic and office/industry uses (perhaps due to some steeper terrains offering view benefits). Estimates of the cross-alternative correlations suggest that a parcel’s residential use “utility” or attractiveness tends to be negatively correlated with that of commercial/civic, but positively associated with that of office/industrial uses, while the latter two land uses exhibit some negative correlation. Using an inverse-distance weight matrix for each parcel’s closest 50 neighbors, the spatial autocorrelation coefficient is estimated to be 0.706, indicating a marked spatial clustering pattern for land development in the selected region.

Appendix

An alternative way to represent the SAR MNP model uses matrix-variate priors. Kadiyala and Karlsson (1997) provided the technicalities of the matrix-variate $t$ and normal distributions, and these are used to guide the derivation of $\rho$’s posterior for the SAR MNP model.

If one writes the stacked latent response as ${\tilde{{\varvec{y}}}^{*}}=({{\varvec{y}}}_\mathbf{1}^{*\prime },\,{{\varvec{y}}}_\mathbf{2}^{*\prime } ,\,\ldots ,\,{{\varvec{y}}}_\mathbf{J}^{*\prime } )$, with ${{\varvec{y}}}_\mathbf{j}^*=(y_{1j}^*,\,\ldots ,\,y_{Nj}^*)^{\prime }$, then ${\tilde{X}}$ should take the form: ${\tilde{X}} =\small {\left[ {{\begin{array}{c@{\quad }c@{\quad }c} X&{} \cdots &{} 0 \\ \vdots &{} \ddots &{} \vdots \\ 0&{} \cdots &{} X \\ \end{array} }} \right] }=I_J \otimes \hbox {X}$, with the $N\times K$ covariate matrix $X$ repeating itself across different alternatives (due to a lack of generic and/or alternative-specific variables). Similarly,$\,{\tilde{W}} =I_J \otimes W\hbox { and } \tilde{\varvec{\varepsilon }}=\small {\left( {{\begin{array}{c} {\varvec{\varepsilon }_1 } \\ {\varvec{\varepsilon }_2 } \\ \vdots \\ {\varvec{\varepsilon }_J } \\ \end{array} }} \right) }$, with $\varvec{\varepsilon }_j =(\varepsilon _{1j} ,\,\ldots ,\,\varepsilon _{Nj} )^{\prime }$. Here, $\tilde{\varvec{\varepsilon }}$ is assumed to follow a $N\left( {\mathbf{0},\,\,\Sigma \otimes I_N } \right)$ distribution, since the correlated error terms of each observation are separate. ${\varvec{\beta }}$ assumes the same form as shown in the body of this paper. This alternative representation offers the possibility of utilizing the matrix-variate distribution, which may facilitate estimation of $\rho$’s posterior distribution, by integrating out the other unknown parameters.

Given the SAR assumption, where ${\tilde{{\varvec{y}}}^{*}}=\rho {\tilde{W}} {\tilde{{\varvec{y}}}^{*}}+{\tilde{X}} {\varvec{\beta }} +\tilde{\varvec{\varepsilon }}$, the likelihood function follows a matrix-variate distribution, with density function $p(\, {{\tilde{{\varvec{y}}}^{*}}|\rho ,\,{\varvec{\beta }} ,\,\Sigma } )=(2\pi )^{-nJ/2}|\Sigma |^{-n/2}\hbox {exp}\{ {-\frac{1}{2}\hbox {tr}[ {\Sigma ^{-1}( {\hbox {A}{\tilde{{\varvec{y}}}^{*}}-X{\varvec{\beta }} } )^{\prime }( {\hbox {A}{\tilde{{\varvec{y}}}^{*}}-X{\varvec{\beta }} } )} ]} \}$, where $A=I_\mathrm{n} -\rho W$.

Assuming a diffuse prior for $\Sigma \hbox { and }{\varvec{\beta }}$, such that $\pi ({\varvec{\beta }} ,\,\Sigma )\propto |\Sigma |^{-\frac{J+1}{2}}$, the joint posterior can be written as follows:

$$\begin{aligned} p\left( {\rho ,\,{\varvec{\beta }} ,\,\Sigma |{\tilde{{\varvec{y}}}^{*}}} \right)&\propto |\Sigma |^{-\frac{J+1}{2}}|\Sigma |^{-\frac{n-k}{2}}\hbox {exp}\left\{ {-\frac{1}{2}\hbox {tr}\left[ {\Sigma ^{-1}\left( {\hbox {A}{\tilde{{\varvec{y}}}^{*}}-X \hat{\varvec{\beta }} } \right) ^{\prime }\left( {\hbox {A}{\tilde{{\varvec{y}}}^{*}}-X \hat{\varvec{\beta }}} \right) } \right] } \right\} \\&\times |\Sigma |^{-\frac{k}{2}}\hbox {exp}\left\{ {-\frac{1}{2}\hbox {tr}\left[ {\Sigma ^{-1}\left( {{\beta } - \hat{\varvec{\beta }} } \right) ^{\prime }X^{\prime }X\left( {{\beta } -\hat{\varvec{\beta }}} \right) } \right] } \right\} \end{aligned}$$

Thus, the conditional posterior for ${\varvec{\beta }}$ follows the matrix-variate normal distribution: $p( {\rho |{\varvec{\beta }} ,\,\Sigma ,\,{\tilde{{\varvec{y}}}^{*}}} )\sim \,MN_{kJ} ( {\hat{\varvec{\beta }} ,\,\Sigma ,\,\,( {X^{\prime }X} )^{-1}} )$.

One advantage of assuming matrix-variate priors lies in the ease in integrating out ${\varvec{\beta }} \hbox { and }\Sigma$, and then obtaining $\rho$’s marginal posterior:

$$\begin{aligned}&p\left( {\rho ,{\varvec{\beta }} |\,{\tilde{{\varvec{y}}}^{*}}} \right) \propto \int p\left( {\rho ,\,{\varvec{\beta }} ,\,\Sigma |{\tilde{{\varvec{y}}}^{*}}} \right) d\Sigma =\left| {S+\left( {{\beta } -\hat{\varvec{\beta }} } \right) ^{\prime }X^{\prime }X\left( {{\beta } - \hat{\varvec{\beta }} } \right) } \right| ^{-\frac{n}{2}}, \hbox { with }\\&S=\left( {\hbox {A}{\tilde{{\varvec{y}}}^{*}}-X \hat{\varvec{\beta }} } \right) ^{\prime } \left( {\hbox {A}{\tilde{{\varvec{y}}}^{*}}-X \hat{\varvec{\beta }} } \right) . \end{aligned}$$

$p\left( {\rho |\,{\tilde{{\varvec{y}}}^{*}}} \right) = \smallint p\left( {\rho ,\,{\varvec{\beta }} |{\tilde{{\varvec{y}}}^{*}}} \right) d{\varvec{\beta }} \propto \,\left| S \right| \,^{-k/2}$ when assuming ${\varvec{\beta }}$ follows matrix-variate $t$ distribution.

One may also consider using the griddy Gibbs sampling technique (LeSage and Pace 2009) to draw directly from the marginal posterior of $\rho$, as shown above.